Question

If $\overline{a}=\overline{i}-2\overline{j}+3\overline{k},\overline{b}=2\overline{i}+\overline{j}+\overline{k},\overline{c}=\overline{i}+\overline{j}+2\overline{k}$ then find $|(\overrightarrow{a}\times \overrightarrow{b})\times \overline{c}|$ and $|\overline{a}\times (\overrightarrow{b}\times \overrightarrow{c})|$.

Answer 1

$\Rightarrow \overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c})=(\overrightarrow{a}\cdot \overrightarrow{c})\cdot \overrightarrow{b}-(\overrightarrow{a}\cdot \overrightarrow{b})\overrightarrow{c}=5\overrightarrow{b}-3\overrightarrow{c}=7\hat{i}+2\hat{j}-\hat{k}$

$(\overrightarrow{a}\times \overrightarrow{b})\times \overrightarrow{c}=(\overrightarrow{a}\cdot \overrightarrow{c})\overrightarrow{b}-(\overrightarrow{b}-\overrightarrow{c})\overrightarrow{a}=5\overrightarrow{b}-5\overrightarrow{a}=5(\hat{i}+3\hat{j}-2\hat{k})$

$\Rightarrow |(\overrightarrow{a}\times \overrightarrow{b})\times \overrightarrow{c}|=5\sqrt{1+9+4}=5\sqrt{14}$

and $|\overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c})|=\sqrt{49+4+1}=\sqrt{54}=3\sqrt{6}$.